Mathematical Tripos Examination for 1878

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กระทู้นี้ผมจะโพสต์โจทย์พร้อมเฉลยของ Mathematical Tripos Examination for 1878 ลักษณะการสอบเป็นดังนี้

It may be convenient to state that the examination occupies nine days, the first four days being separated from the last five days by an interval of ten days. The examination began on Monday, December 31, 1877, and ended on Friday, January 18, 1878.


การสอบใน 3 วันแรกมีเนื้อหาที่ต้องอ่านเตรียมสอบดังนี้

The subjects of examination on the first three days (Monday, December 31, 1877, to Wednesday,
January 2, 1878) were the following:

$\bullet$ Euclid. Books I. to VI. Book XI. Props, i. to xxi. Book XII. Props, i., ii.
$\bullet$ Arithmetic; and the elementary parts of Algebra; namely, the rules for the fundamental operations upon algebraical symbols with their proofs, the solution of simple and quadratic equations, ratio and proportion, arithmetical, geometrical and harmonical progression, permutations and combinations, the binomial theorem and logarithms.
$\bullet$ The elementary parts of Plane Trigonometry, so far as to include the solution and properties of triangles.
$\bullet$ The elementary parts of Conic Sections, treated geometrically, but not excluding the method of orthogonal projections ; curvature.
$\bullet$ The elementary parts of Statics; namely, the equilibrium of forces acting in one plane and of parallel forces, the centre of gravity, the mechanical powers, friction.
$\bullet$ The elementary parts of Dynamics; namely, uniform, uniformly accelerated, and uniform circular motion, falling bodies and projectiles in vacuo, cycloidal oscillations, collisions, work.
$\bullet$ The first, second, and third sections of Newton's Principia; the propositions to be proved by Newton's methods.
The elementary parts of Hydrostatics; namely, the pressure of fluids, specific gravities, floating bodies, density of gases as depending on pressure and temperature, the construction and use of the more simple instruments and machines.
$\bullet$ The elementary parts of Optics; namely, the reflexion and refraction of light at plane and spherical surfaces, not including aberrations ; the eye; construction and use of the more simple instruments.
$\bullet$ The elementary parts of Astronomy ; so far as they are necessary for the explanation of the more simple phenomena, without the use of spherical trigonometry; astronomical instruments.


เนื้อหาเยอะมากเลยใช่ไหมครับ ... แต่ 4 วันหลังยิ่งแบ่งเนื้อหายุ่งไปอีกตามข้อความข้างล่างนี้

The subjects of examination on the fourth day (Thursday, January 3, 1878) and on the last five days
(Monday, January 14, 1878 to Friday, January 18, 1878) are contained in the following schedule of
the divisions of the subjects.

First Division.
$\bullet$ Algebra.
$\bullet$ Trigonometry; Plane and Spherical.
$\bullet$ Theory of Equations.
$\bullet$ Analytical Geometry ; Plane and Solid.
$\bullet$ Finite Differences.
$\bullet$ Differential and Integral Calculus.
$\bullet$ Differential Equations.
$\bullet$ Statics.
$\bullet$ Hydrostatics.
$\bullet$ Dynamics of a Particle.
$\bullet$ Dynamics of Rigid Bodies.
$\bullet$ Optics.
$\bullet$ Spherical Astronomy.

Second Division.
$\bullet$ Higher parts of Algebra and of the Theory of Equations.
$\bullet$ Higher parts of Finite Differences.
$\bullet$ Elliptic Functions.
$\bullet$ Higher parts of Analytical Geometry.
$\bullet$ Higher parts of Differential Equations.
$\bullet$ Calculus of Variations.
$\bullet$ Theory of Chances, including combination of observations.

Third Division.
$\bullet$ Newton's Principia, Book I., Sections ix. and XI.
$\bullet$ Lunar and Planetary Theories.
$\bullet$ Higher Parts of Dynamics.
$\bullet$ Laplace's Coefficients.
$\bullet$ Attractions.
$\bullet$ Figure of the Earth.
$\bullet$ Precession and Nutation.

Fourth Division.
$\bullet$ Hydrodynamics.
$\bullet$ Theory of Sound.
$\bullet$ Physical Optics.
$\bullet$ Waves and Tides.
$\bullet$ Vibrations of Strings and Bars.
$\bullet$ Theory of Elastic Solids treated as continuous.

Fifth Division.
$\bullet$ Expression of arbitrary functions by series or integrals involving sines or cosines.
$\bullet$ Heat.
$\bullet$ Electricity.
$\bullet$ Magnetism.


ตารางการแบ่งเนื้อหาที่ใช้สอบในแต่ละวันเป็นดังนี้

The Examination was conducted according to the following schedule.

Monday, Dec. 31, 1877
09:00-12:00 $\;\;\;$ Euclid and Conics.
13:30-16:00 $\;\;\;$ Arith. Algebra and Plane Trigonometry.

Teusday, Jan. 1, 1878
09:00-12:00 $\;\;\;$ Statics and Dynamics.
13:30-16:00 $\;\;\;$ Hydrostatics and Optics.

Wednesday, Jan. 2, 1878
09:00-12:00 $\;\;\;$ Easy Problem in all the three days' Subjects.
13:30-16:00 $\;\;\;$ Newton and Astronomy.

Thursday, Jan. 3, 1878
09:00-12:00 $\;\;\;$ Easy questions from 1st Div. and from Phys. Sub. in other Division.
13:30-16:00 $\;\;\;$ Easy questions from 1st Div. and from Phys. Sub. in other Division.

Monday, Jan. 14, 1878
09:00-12:00 $\;\;\;$ Natural Philos. from 1st Div.
13:30-16:00 $\;\;\;$ Pure Math. from 1st Div.

Teusday, Jan. 15, 1878
09:00-12:00 $\;\;\;$ Easy problems.
13:30-16:00 $\;\;\;$ 1st Division.

Wednesday, Jan. 16, 1878
09:00-12:00 $\;\;\;$ Easy problems.
13:30-16:00 $\;\;\;$ 2nd, 4th, 5th Division.

Thursday, Jan. 17, 1878
09:00-12:00 $\;\;\;$ 2nd, 3rd, 4th Division.
13:30-16:00 $\;\;\;$ 2nd, 3rd, 5th Division.

Friday, Jan. 18, 1878
09:00-12:00 $\;\;\;$ 2nd, 3rd, 4th, 5th Division.
13:30-16:00 $\;\;\;$ 3rd, 4th, 5th Division.

ผู้รักคณิตศาสตร์ทุกคนคงไม่อยากพลาดชมข้อสอบชุดนี้แน่
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MONDAY, December 31, 1877. 9:00 to 12:00

1. Parallelograms on the same base, and between the same parallels, are equal to one another.
$\;\;\;$ In a given triangle it is required to inscribe a parallelogram equal to half the triangle, so that one side is in the same straight line with one side of the triangle and has one extremity at a given point of that side.

2. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected and of the square on the line made up of the half and the part produced.
$\;\;\;$ If a straight line $AB$ be bisected in $C$ and produced to $D$ so that the square on $AD$ is three times the square on $CD$, and if $GB$ be bisected in $E$, shew that the square on $ED$ is three times the square on $EB$.

3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
$\;\;\;$ If a quadrilateral be inscribed in a circle, and the middle points of the arcs subtended by its sides be joined to make another quadrilateral, and so on : shew that these quadrilaterals tend to become squares.

4. Inscribe a circle in a given triangle.
$\;\;\;$ Prove that four circles may be described, touching the three sides of a triangle, and that the square on the distance between the centres of any two, together with the square on the distance between the centres of the other two, is equal to the square on the diameter of the circle passing through the centres of any three.

5. If the sides of two triangles, about each of their angles, be proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides.
$\;\;\;$ The side $BC$ of a triangle $ABC$ is produced to $D$, so that the triangles $ABD, ACD$ are similar. Prove that $AD$ touches the circle described about the triangle $ABC$.

6. If two straight lines be cut by parallel planes, they shall be cut in the same ratio.
$\;\;\;$ If an equifacial tetrahedron be cut by a plane parallel to two edges which do not meet, the perimeter of the parallelogram in which it is cut shall be double of either edge of the tetrahedron to which it is parallel.

7. In the parabola, prove that the distance between the foot of the ordinate of any point and the foot of the normal at that point is equal to half the latus-rectum.
$\;\;\;$ Inscribe a circle in the segment of a parabola cut off by a double ordinate.

8. Prove that the tangent, at any point of an ellipse, makes equal angles with the focal distances of that point.
$\;\;\;$ From the foci of an ellipse, perpendiculars are let fall to the tangent at any given point of the curve. With the feet of these perpendiculars as foci, an ellipse is described touching the axis major of the given ellipse. Prove that the point, in which it touches the axis major, will be the foot of the ordinate of the given point, and that the ellipse described will be similar to the given ellipse.

9. Define conjugate diameters of an ellipse ; and prove that the sum of the squares on any two conjugate diameters is constant.
$\;\;\;$ A system of parallelograms is inscribed in an ellipse whose sides are parallel to the equi-conjugate diameters ; prove that the sum of the squares on the sides is constant.

10. The tangents drawn from any point to a conic subtend equal angles at the focus.
$\;\;\;$ If $P$ be any point of an hyperbola whose foci are $S$ and $H$, and if the tangent at $P$ meet an asymptote in $T$, the angle between that asymptote and $HP$ is double the angle $STP$.

11. If two chords of a rectangular hyperbola be at right angles, each of their four extremities is the orthocentre of the triangle formed by the other three.
$\;\;\;$ If four such points and the tangent at one of them be given, find the centre of the hyperbola.

12. The section of a cone by a plane, which is not perpendicular to the axis and does not pass through the vertex, is either an ellipse, a parabola, or an hyperbola.
$\;\;\;$ In a given plane are drawn a series of confocal conies, upon which stand cones with their vertical angles right angles. Shew that the locus of their vertices is given by the intersection of an hyperbola, whose vertices are the foci of the conies, and a circle concentric with the hyperbola and passing through its foci.

ครบแล้วครับสำหรับข้อสอบชุดแรก
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ใครคิดข้อไหนออก ขอเชิญโพสต์วิธีทำได้เลยครับ ผมอยากเห็นแนวคิดหลายๆ แบบ

ผมอยากรู้ด้วยว่าแต่ละคนอ่านโจทย์แล้ว คิดว่าข้อสอบชุดนี้ยากง่ายแค่ไหน เพราะสำหรับผมมันยากกกส์

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ข้อสอบโหดมากครับ เฉลี่ยข้อละสิบห้านาที ในขณะที่บางข้อผมคิดหนึ่งชั่วโมงก็อาจจะยังคิดไม่ออก

ยังไม่ได้ทด แต่จะลงวิธีทำให้ถ้ามีเวลาครับ

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ว่ากันว่าพวกที่สอบได้คะแนนสูงสุด (Senior Wrangler) หรือพวกอันดับต้นๆ นั้น มีอยู่หลายคนที่ว่าพอสอบเสร็จ
วันสุดท้ายแล้ว ลุกขึ้นเดินได้ไม่เกิน 50 หลา และหลังจากนั้นต้องนอนพักฟื้นนานถึง 3 เดือน

นี่คือข้อความที่ผมอ่านเจอในบทความบางฉบับ

Tripos stress:
Cakewalk (เหยียบหิมะไร้รอย) : .......... Cayley(SW, 1842), Rayleigh(SW, 1865)
Had enough (สีทนได้) : ................. Kelvin (2W, 1845)
Nerves (มึน-เครียด-สอบต่อ) : ........... Maxwell (2W, 1854), J.J. Thomson(2W, 1880)
Nervous breakdown (หัวระเบิด) : ...... James Wilson (SW, 1859)

Could not walk 50 yards afterwards Took 3 months to recuperate & only by forgetting
all Cambridge mathematics

สำหรับ Cayley ผู้ทำข้อสอบ Tripos ได้แบบฉลุย (Cakewalk) เขียนบทความคณิตศาสตร์ตลอดชีวิตไว้มากถึง 947 ชิ้น
(ทำลายสถิติของ Euler ซึ่งเขียนบทความไว้รวม 800 ชิ้น)

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มาดูกันต่อครับว่า ข้อสอบของบ่ายวันแรกจะเป็นอย่างไรบ้าง จะมีส่วนให้อาหารเที่ยงย่อยง่ายขึ้นหรือแทบสำลัก


MONDAY, December 31, 1877. 13:30 to 16:00 (เขาระบุไว้ 2.5 ชั่วโมงจริงๆ ?)

1. Obtain the value of $\pi$ to six places of decimals from the series
$$\pi = \frac{14}{5}\left[1 + \frac23\left(\frac{1}{50}\right) + \frac{2\cdot4}{3\cdot5}\left(\frac{1}{50}\right)^2 + \frac{2\cdot4\cdot6}{3\cdot5\cdot7}\left(\frac{1}{50}\right)^3+...\right]$$
$$\;\;\;+ \frac{948}{3125}\left[1 + \frac23\left(\frac{9}{6250}\right) + \frac{2\cdot4}{3\cdot5}\left(\frac{9}{6250}\right)^2 + \frac{2\cdot4\cdot6}{3\cdot5\cdot7}\left(\frac{9}{6250}\right)^3+...\right]$$
$\;\;\;$ If $\frac{p}{n}$ and $\frac{n-p}{n}$ be converted into circulating decimals, find the relation between the figures in their periods, $n$ being supposed to be prime to 10 and $p$ less than $n$.

2. Prove that, if $\phi(x)$ be a rational and integral function of $x$ which vanishes when $x$ is put equal to $a$, then $x-a$ is a factor of $\phi(x)$
$\;\;\;$ Shew that $b^2(a-b)(c-b)\left[(a-b)^2+(c-b)^2\right]-ab^2c(a^2+c^2)+b^5(a-b+c)$ is a complete cube.

3. Explain to what extent the equation $a^m \cdot a^n = a^{m+n}$ admits of being proved.
$\;\;\;$ Obtain the values of $a^0$ and $a^{-\frac12}$.
$\;\;\;$ If $\phi(x) = \frac{a^x-a^{-x}}{a^x+a^{-x}}, \;\; F(x) = \frac{2}{a^x+a^{-x}},$ shew that
$$\phi(x+y) = \frac{\phi(x)+\phi(y)}{1+\phi(x)\phi(y)}, \;\; F(x+y) = \frac{F(x)F(y)}{1+\phi(x)\phi(y)}.$$

4. Find the sum of the cubes of the first $n$ natural numbers ; and shew that every cube is the difference of two squares, such that if the cube contains an uneven factor $a^3$, each of the squares is divisible by $a^2$.
$\;\;\;$ Find the sum of the cubes of $n$ consecutive terms of an arithmetical progression ; and shew that it is divisible by the sum of the corresponding $n$ terms of the arithmetical progression.

5. Solve the equations :
$$ x^2-(2a-b-c)x + a^2 +b^2 +c^2-bc-ca-ab = 0;$$
$$ x^2+2xy-y^2 = ax+by,\;\;\; x^2-2xy-y^2 = bx-ay$$
$\;\;\;$ If $x^2 = px+q,$ shew that
$$ x^n = \frac{\alpha^n-\beta^n}{\alpha-\beta}x + q\frac{\alpha^{n-1}-\beta^{n-1}}{\alpha-\beta},$$
where $\;\;\; \alpha+\beta = p, \;\; \alpha \beta = -q.$
$\;\;\;$ If $x^3 = px^2 + qx+r,$ express $x^n$ in the form $Ax^2+Bx+C.$

6. Prove that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities.
$\;\;\;$ Given that $a^2 + b^2 = 1$ and that
$\;\;\;\;\;\; log 2 = 0.3010300, \;\;log (1 + a) = 0.1928998, \;\;log (1+b) = 0.2622226,$
$\;\;\;$ find $log(1 + a + b).$

7. Define the Trigonometrical Ratios of an angle so as to be true for all magnitudes of the angle ; and make a table shewing the values of the trigonometrical ratios in terms of any one of them.
$\;\;\;$ Prove that the equation $\tan x = kx$ has an infinite number of real roots.

8. Prove geometrically
$\;\;\; (i) \;\;\; \sin A+\sin B = 2\sin\frac12(A+B)\cos\frac12(A-B),$
$\;\;\; (ii) \;\; \cos A-\cos B = 2\sin\frac12(B+A)\sin\frac12(B-A);$
and explain how such formulae are shewn to be true universally for all magnitudes of the angles $A$ and $B$.
If $$\;\;\; \frac{\cos(\alpha+\beta+\theta)}{\sin(\alpha+\beta)\cos^2\gamma} = \frac{\cos(\gamma+\alpha+\theta)}{\sin(\gamma+\alpha)\cos^2\beta},$$
and $\beta$ and $\gamma$ are unequal, prove that each member will be equal to
$$\frac{\cos(\beta+\gamma+\theta)}{\sin(\beta+\gamma)\cos^2\alpha},$$
and that
$$ \cot\theta = \frac{\sin(\beta+\gamma)\sin(\gamma+\alpha)\sin(\alpha+\beta)}{\cos(\beta+\gamma)\cos(\gamma+\alpha)\cos(\alpha+\beta) + \sin^2 (\alpha + \beta + \gamma)}.$$

9. Find an expression for all angles having a given sine.
$\;\;\;$ If $A+B+C+D = 2\pi,$ then
$$ \cos\frac12A \cos\frac12D \sin\frac12B \sin\frac12C - \cos\frac12B \cos\frac12C \sin\frac12A \sin\frac12D $$
$$\;\;\; = \sin\frac12(A + B) \sin\frac12(A + C) \cos\frac12(A +D).$$

10. Prove that, in any triangle, $a^2 = b^2+c^2-2bc \cos A;$ and hence prove that
$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$$
$\;\;\;$ In a triangle $ABC, AB = AC + \frac12BC,$ and $BC$ is divided in $O$ so that $BO : OC = 1 : 3;$ prove that the angle $ACO$ is twice the angle $AOC.$

11. Solve a triangle in which one angle at the base, the opposite side, and the altitude, are given ; and explain when the solution is ambiguous or impossible.
$\;\;\;$ If the angle is $36^\circ,$ the opposite side $4,$ and the altitude $\sqrt5 - 1,$ solve the triangle.

12. Find expressions for the radii of the inscribed and circumscribed circles of a triangle.
$\;\;\;$ If the centre of the inscribed circle of a triangle be equidistant from the centre of the circumscribed circle and the orthocentre, prove that one angle of the triangle is $60^\circ$.


ข้อสอบชุดนี้ครบทั้ง 12 ข้อแล้วครับ ... เชิญผู้สนใจลุยโจทย์ตามสบาย
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เป็นยังไงกันบ้างครับ พอได้เห็นข้อสอบของบ่ายวันแรกแล้ว อาหารเที่ยงย่อยง่ายขึ้นหรือเปล่า ?

สำหรับผมเอง ... ท้อไปตั้งแต่ข้อแรกๆ ของช่วงเช้าแล้ว
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กำเนิดของคำว่า Tripos

อาจมีบางคนอย่างรู้ที่มาของคำว่า Tripos อย่างละเอียดและถูกต้อง ซึ่งหลายคนอาจหาอ่านได้จากพวก Wikipedia อยู่แล้ว แต่ว่ายังมีแหล่งข้อมูลบางแห่งที่ค้นหาในเน็ตได้ยาก อย่างเช่น ข้อมูลในหนังสือยุคเก่าที่กล่าวถึงประวัติของ Tripos เป็นต้น

ข้อความต่อไปนี้ผมคัดจากหนังสือ A History of the Study of Mathematics at Cambridge แต่งโดย W.W. Rouse Ball (Fellow and Lecturer of Trinity College, Cambridge; Author of A History of Mathematics) เมื่อปี 1889 ซึ่งน่าจะให้ข้อมูลได้ถูกต้องชัดเจนกว่าแหล่งข้อมูลอื่น

$\;\;\;$ The curious origin of the term tripos has been repeatedly told, and an account of it may fitly close this chapter. There were three principal occasions on which questionists were admitted to the degree of bachelor. The first of these was the comitia priora held on Ash-Wednesday for the best men in the year. The next was the comitia posteriora which was held a few weeks later, and at which any student who had distinguished himself in the quadragesimal exercises subsequent to Ash-Wednesday had his seniority reserved to him. Lastly, there was the comitia minora, or the general bachelor's commencement, for students who had in no special way distinguished themselves. In the fifteenth century an important part in the ceremony on each of these occasions was taken by a certain "ould bachilour," who as the representative of the university had to sit upon a three-legged stool or tripos "before Mr Proctours" and test the abilities of the would-be graduates by arguing some question with the "eldest son," who was the senior and representative of them. To assist the latter in what was generally an unequal contest, his "father," that is, the officer of his college who was to present him for his degree, was allowed to come to his assistance.

$\;\;\;$ The ceremony was a serious one, and had a certain religious character. It took place in Great St Mary's Church, and marked the admission of the student to a position with new responsibilities, while the season of Lent 1 was chosen with a view to bring this into prominence. The puritan party objected to the observance of such ecclesiastical ceremonies, and in the course of the sixteenth century they converted the proceedings into a sort of licensed buffoonery. The part played by the questionist became purely formal. A serious debate still sometimes took place between the father of the senior questionist and a regent master, who represented the university; but the discussion always began with an introductory speech by the bachelor, who came to be called Mr Tripos just as we speak of a judge as the bench or of a rower as an oar. Ultimately the tripos was allowed to say pretty much what he pleased, so long as it was not dull and was scandalous. The speeches he delivered or the verses he recited were generally preserved by the registrary, and were known as the tripos verses : originally they referred to the subjects of the disputations then propounded. The earliest copies now extant are those for 1575.

$\;\;\;$ The university officials, to whom the personal criticisms in which the tripos indulged were by no means pleasing, repeatedly exhorted him to remember "while exercising his privilege of humour, to be modest withal." In 1740, says Mr Mullinger, "the authorities after condemning the excessive license of the tripos announced that the comitia at Lent would in future be conducted in the senate-house ; and all members of the university, of whatever order or degree, were forbidden to assail or mock the disputants with scurrilous jokes or unseemly witticisms. About the year 1747-8, the moderators initiated the practice of printing the honour lists on the back of the sheets containing the tripos-verses, and after the year 1755 this became the invariable practice. By virtue of this purely arbitrary connection these lists themselves became known as the tripos; and eventually the examination itself, of which they represented the results, also became known by the same designation."

$\;\;\;$ A somewhat similar position at the comitia majora (or congregation held on Commencement-day) to that of the tripos on Ash-Wednesday was filled by the prevaricator or varier, who was the junior M.A. regent of the previous year, or his proxy. But he never indulged in as much license as the "ould bachilor," and no determined effort to turn that ceremony into a farce was ever made.

$\;\;\;$ The tripos and prevaricator ceased to recite their speeches about 1750, but the issue of the verses by the former has never been discontinued. At present these verses are published on the last day of the Michaelmas term, and consist of four odes, usually in Latin but occasionally in Greek, in which current events or topics of conversation in the university are treated satirically or seriously. They are written for the two proctors and two moderators by undergraduates or commencing bachelors, who are supposed each to receive a pair of white kid gloves in recognition of their labours. Since 1859 the two sets, corresponding to the two days of admission, have been printed together on the first three pages of a sheet of foolscap paper. On the fourth page the order of seniority of the honour men of the year is printed crosswise in columns, the sheet being folded into four parts, so that all the names can be read without opening the page to more than half its extent.

$\;\;\;$ Thus gradually the word tripos changed its meaning "from a thing of wood to a man, from a man to a speech, from a speech to two sets of verses, from verses to a sheet of coarse foolscap paper, from a paper to a list of names, and from a list of names to a system of examination."


ใครมีฝีมือแปลเจ๋งๆ ช่วยแปลให้เพื่อนคนอื่นอ่านด้วยก็จะดีครับ เพราะว่ากำเนิดของคำยาวน่าดูเลย
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ผมโพสต์กำเนิดของคำว่า Tripos ไว้ทั้ง 2 กระทู้ที่ผมตั้งไว้ตอนนี้ เผื่อว่านานไป 2 กระทู้อาจไม่อยู่เกาะติดกันอย่างตอนเริ่มต้น
ถึงตอนนั้นใครเปิดอ่านกระทู้ใดกระทู้หนึ่งก็จะได้รู้ข้อมูลนี้เสมอ

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ผมเพิ่มคำอธิบายเกี่ยวกับเนื้อหาที่ต้องใช้ในการสอบ Tripos ไว้ในข้อความที่ใช้ตั้งกระทู้แล้ว เพื่อนๆ จะได้รู้ว่ามีอะไรบ้าง
ที่ต้องอ่านเตรียมสอบกันในยุคนั้น ซึ่งอ่านแล้วจะเห็นว่าเป็นการสอบแบบรวบยอดทุกวิชาที่เรียนมาในหลายปีก็ว่าได้

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มีข้อสอบของบ่ายวันแรกเพิ่มเติมแล้วในความเห็นที่ 6 เชิญทุกคนลองแก้โจทย์ดู ว่าจะคิดออกหรือเปล่า ?
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ตอนนี้ข้อสอบของบ่ายวันแรกในความเห็นที่ 6 ครบทั้ง 12 ข้อแล้ว ... พิมพ์สมการเหนื่อยน่าดู

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มาลองดูเฉลยของข้อสอบชุดแรกกันดีกว่า ... ใครเก่งภาษาอังกฤษ รบกวนช่วยแปลให้คนที่ไม่แข็งภาษาอ่านด้วยครับ


เฉลยข้อสอบชุดที่ 1: MONDAY, December 31, 1877. 9:00 to 12:00

1. Parallelograms on the same base, and between the same parallels, are equal to one another.
$\;\;\;$ In a given triangle it is required to inscribe a parallelogram equal to half the triangle, so that one side is in the same straight line with one side of the triangle and has one extremity at a given point of that side.

Solution:

$\;\;\;$ Let $ABC$ be the given triangle ; $D$ the given point in $BC$ ; and let $BD < DC.$
$\;\;\;$ Bisect $AB, AC$ in $E, F$. Join $EF, ED$. Draw $FG$ parallel to $ED$ to meet $BC$ in $G$. $EDGF$ is the required parallelogram.
$\;\;\;$ Join $EC, FB$. Then $\Delta AEF= \Delta BEF,$ these being on equal bases; similarly $\Delta AFE= \Delta CFE$; therefore $\Delta BEF= \Delta CFE;$ therefore $EF, BC$ are parallel ; therefore, by construction, $EDGF$ is a parallelogram, and $= 2 \Delta EBF = \Delta ABF = \frac12 \Delta ABC.$

อาจจะเข้าใจยากซักหน่อย เพราะว่าทั้งเล่มที่ผมได้มานั้น ไม่มีรูปประกอบเลย ทั้งที่มีการอ้างถึงแต่คงพิมพ์ตกไปหมด
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เฉลยข้อสอบชุดที่ 1: MONDAY, December 31, 1877. 9:00 to 12:00

2. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected and of the square on the line made up of the half and the part produced.
$\;\;\;$ If a straight line $AB$ be bisected in $C$ and produced to $D$ so that the square on $AD$ is three times the square on $CD$, and if $GB$ be bisected in $E$, shew that the square on $ED$ is three times the square on $EB$.

Solution:

We have, by the proposition,
$\;\;\;$ sq. on $AD +$ sq. on $BD = 2$ sq. on $AC + 2$ sq. on $CD$;
and by hypothesis, $3$ sq. on $CD =$ sq. on $AD$.

Adding these equals to the above equals, and then taking from each sum the sq. on $AD + 2$ sq. on $CD$, we have
$\;\;\;$ sq. on $CD +$ sq. on $BD = 2$ sq. on $AC = 2$ sq. on $CB = 8$ sq. on $CE$.
But by the proposition,
$\;\;\;$ sq. on $CD +$ sq. on $BD = 2$ sq. on $CE + 2$ sq. on $ED$,
therefore $\;\;\; 2$ sq. on $CE + 2$ sq. on $ED = 8$ sq. on $CE$,
therefore $\;\;\; 2$ sq. on $ED = 6$ sq. on $CE$,
therefore $\;\;\;$ sq. on $ED = 3$ sq. on $CE = 3$ sq. on $EB$.


เป็นไงบ้างครับ เฉลยที่ให้นี้ยากหรือง่ายกว่าวิธีที่คุณลองคิดเอง
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เฉลยข้อสอบชุดที่ 1: MONDAY, December 31, 1877. 9:00 to 12:00

3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
$\;\;\;$ If a quadrilateral be inscribed in a circle, and the middle points of the arcs subtended by its sides be joined to make another quadrilateral, and so on : shew that these quadrilaterals tend to become squares.

Solution:

$\;\;\;$ Let $A_1B_1C_1D_1, \;A_2B_2C_2D_2, \;A_3B_3C_3D_3$ be three successive quadrilaterals, made as the question directs, the order of points being given by

$A_1A_2B_1B_2......\;$ and $\;A_2A_3B_2B_3......$

Then

$2$ arc $A_2B_2 =$ arc $A_1B_1 +$ arc $B_1C_1,$
$2$ arc $B_2C_2 =$ arc $B_1C_1 +$ arc $C_1D_1,$
$............ = ............ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

$\;\;\;$ Let $A_1B_1$ be the greatest; $B_1C_1$ or $C_1D_1$ the least of the arcs $A_1B_1,\; B_1C_1,\; C_1D_1,\; D_1A_1$. Then the above equalities shew that the arc $A_2B_2$ is less than the greatest and greater than the least of these arcs, and that the same is true for the arcs $B_2C_2,\; C_2D_2,\; D_2A_2$. Hence $A_3B_3C_3D_3$ has no side so great and no side so small as the greatest and least sides of $A_1B_1C_1D_1$ respectively: i.e. $A_2B_2C_2D_2$ is more nearly equilateral than $A_1B_1C_1D_1$: and so on. Hence the quadrilaterals tend to become rhombi, but the only rhombus that can be inscribed in a circle is a square, therefore the quadrilaterals tend to become squares.

ความจริงข้อนี้มีเฉลยอีกแบบหนึ่ง แต่ผมโพสต์ให้แค่แบบเดียวก่อน เผื่อว่าจะมีใครคิดและโพสต์อีกแนวคิดหนึ่งเพิ่มเติมได้

โดยส่วนตัวแล้ว ผมชอบคำถามของโจทย์ของข้อนี้มาก ชื่นชมว่าคนตั้งโจทย์เขาช่างคิดดี
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